This book investigates Aristotle’s views on abstraction and explores how he uses it. In this work, the author follows Aristotle in focusing on the scientific detail first and then approaches the metaphysical claims, and so creates a reconstructed theory that explains many puzzles of Aristotle’s thought. Understanding the details of his theory of relations and abstraction further illuminates his theory of universals. Some of the features of Aristotle’s theory of abstraction developed in this book include: abstraction is a relation; perception (...) and knowledge are types of abstraction; the objects generated by abstractions are relata which can serve as subjects in their own right, whereupon they can appear as items in other categories. The author goes on to look at how Aristotle distinguishes the concrete from the abstract paronym, how induction is a type of abstraction which typically moves from the perceived individuals to universals and how Aristotle’s metaphysical vocabulary is "relational.’ Beyond those features, this work also looks at how of universals, accidents, forms, causes and potentialities have being only as abstract aspects of individual substances. An individual substance is identical to its essence; the essence has universal features but is the singularity making the individual substance what it is. These theories are expounded within this book. One main attraction in working out the details of Aristotle’s views on abstraction lies in understanding his metaphysics of universals as abstract objects. This work reclaims past ground as the main philosophical tradition of abstraction has been ignored in recent times. It gives a modern version of the medieval doctrine of the threefold distinction of essence, made famous by the Islamic philosopher, Avicenna. (shrink)

Joe Sachs has followed up his brilliant translation of Aristotle's Physics with a new translation of Metaphysics. Sachs's translations bring distinguished new light onto Aristotle's works, which are foundational to history of science. Sachs translates Aristotle with an authenticity that was lost when Aristotle was translated into Latin and abstract Latin words came to stand for concepts Aristotle expressed with phrases in everyday Greek language. When the works began being translated into English, those abstract Latin words or their cognates were (...) used, thus suggesting a level of jargon and abstraction, and in some cases misleading interpretation, which was not Aristotle's language or style. These important new translations open up Aristotle's original thought to readers. (shrink)

This paper is a contribution to our understanding of the constructive nature of Greek geometry. By studying the role of constructive processes in Theodoius’s Spherics, we uncover a difference in the function of constructions and problems in the deductive framework of Greek mathematics. In particular, we show that geometric problems originated in the practical issues involved in actually making diagrams, whereas constructions are abstractions of these processes that are used to introduce objects not given at the outset, so that their (...) properties can be used in the argument. We conclude by discussing, more generally, ancient Greek interests in the practical methods of producing diagrams. (shrink)

Stephen Makin presents a clear and accurate new translation of an influential and much-discussed part of Aristotle's philosophical system, accompanied by an analytical and critical commentary focusing on philosophical issues. In Book Theta of the Metaphysics Aristotle introduces the concepts of actuality and potentiality---which were to remain central to philosophical analysis into the modern era---and explores the distinction between the actual and the potential.

An Aristotelian Philosophy of Mathematics breaks the impasse between Platonist and nominalist views of mathematics. Neither a study of abstract objects nor a mere language or logic, mathematics is a science of real aspects of the world as much as biology is. For the first time, a philosophy of mathematics puts applied mathematics at the centre. Quantitative aspects of the world such as ratios of heights, and structural ones such as symmetry and continuity, are parts of the physical world and (...) are objects of mathematics. Though some mathematical structures such as infinities may be too big to be realized in fact, all of them are capable of being realized. Informed by the author's background in both philosophy and mathematics, but keeping to simple examples, the book shows how infant perception of patterns is extended by visualization and proof to the vast edifice of modern pure and applied mathematical knowledge. (shrink)

An examination of the emergence of the phenomenon of deductive argument in classical Greek mathematics.

Aristotle says in the De Sensu that other colours are produced through the mixture of black bodies with white . The obvious mixture for him to be referring to is the mixture of the four elements, earth, air, fire, and water, which he describes at such length in the De Generatione et Corruptione. All compound bodies are produced ultimately through the mixture of these elements. The way in which the elements mix is described in i. 10 and 2. 7. They (...) mix in such a way as to produce an entirely new substance, in which the characteristics of the original earth, air, fire, and water survive only in modified form. (shrink)

I offer a new interpretation of Aristotle's philosophy of geometry, which he presents in greatest detail in Metaphysics M 3. On my interpretation, Aristotle holds that the points, lines, planes, and solids of geometry belong to the sensible realm, but not in a straightforward way. Rather, by considering Aristotle's second attempt to solve Zeno's Runner Paradox in Book VIII of the Physics , I explain how such objects exist in the sensibles in a special way. I conclude by considering the (...) passages that lead Jonathan Lear to his fictionalist reading of Met . M3,1 and I argue that Aristotle is here describing useful heuristics for the teaching of geometry; he is not pronouncing on the meaning of mathematical talk. (shrink)

Whether aristotle wrote a work on mathematics as he did on physics is not known, and sources differ. this book attempts to present the main features of aristotle's philosophy of mathematics. methodologically, the presentation is based on aristotle's "posterior analytics", which discusses the nature of scientific knowledge and procedure. concerning aristotle's views on mathematics in particular, they are presented with the support of numerous references to his extant works. his criticism of his predecessors is added at the end.

When ancient mathematical treatises lack expositions of numerical techniques, what purposes could ancient mathematical theories be expected to serve? Ancient writers only rarely address questions of this sort directly. Possible answers are suggested by surveying geometry, mechanics, optics, and spherics to discover how the mathematical treatments imply positions on this issue. This survey shows the ways in which these ancient theoretical inquiries reflect practical activity in their fields. This account, in turn, suggests that the authors may have intended their theorems (...) not to predict, but to explain phenomena. We may then consider what kind of explanations they were seeking. (shrink)

Aristotle’s philosophy of geometry is widely interpreted as a reaction against a Platonic realist conception of mathematics. Here I argue to the contrary that Aristotle is concerned primarily with the methodological question of how universal inferences are warranted by particular geometrical constructions. His answer hinges on the concept of abstraction, an operation of “taking away” certain features of material particulars that makes perspicuous universal relations among magnitudes. On my reading, abstraction is a diagrammatic procedure for Aristotle, and it is through (...) imagined variation of diagrams that one can universalize spatial inferences. In order to shift the discussion of Aristotle’s work on geometry from ontological to methodological questions, I argue in the first section that geometrical properties for Aristotle are necessary accidents of the particulars, canonically diagrams, in which they inhere. I then consider a line of research that shows how diagrams in ancient geometry are not illustrations of the textual proof but are in part constitutive of geometrical inference, an insight I claim can shed light on Aristotle’s philosophical project. Next I substantiate this assertion by giving an alternative interpretation of abstraction according to which it is a diagrammatic procedure that allows a part of a particular diagram to stand in for a universal. In the last section, I consider Aristotle’s account of mathematical cognition, arguing that there is evidence for the view that imagination is necessary both for the construction of figures, and for their presentation as abstractions subject to universal inference. (shrink)

Originally published in 1949. This meticulously researched book presents a comprehensive outline and discussion of Aristotle’s mathematics with the author's translations of the greek. To Aristotle, mathematics was one of the three theoretical sciences, the others being theology and the philosophy of nature. Arranged thematically, this book considers his thinking in relation to the other sciences and looks into such specifics as squaring of the circle, syllogism, parallels, incommensurability of the diagonal, angles, universal proof, gnomons, infinity, agelessness of the universe, (...) surface of water, meteorology, metaphysics and mechanics such as levers, rudders, wedges, wheels and inertia. The last few short chapters address ‘problems’ that Aristotle posed but couldn’t answer, related ethics issues and a summary of some short treatises that only briefly touch on mathematics. (shrink)

Journal Name: Apeiron Issue: Ahead of print.

Both literalism, the view that mathematical objects simply exist in the empirical world, and fictionalism, the view that mathematical objects do not exist but are rather harmless fictions, have been both ascribed to Aristotle. The ascription of literalism to Aristotle, however, commits Aristotle to the unattractive view that mathematics studies but a small fragment of the physical world; and there is evidence that Aristotle would deny the literalist position that mathematical objects are perceivable. The ascription of fictionalism also faces a (...) difficult challenge: there is evidence that Aristotle would deny the fictionalist position that mathematics is false. I argue that, in Aristotle's view, the fiction of mathematics is not to treat what does not exist as if existing but to treat mathematical objects with an ontological status they lack. This form of fictionalism is consistent with holding that mathematics is true. (shrink)

The Regress: Knowledge, we like to suppose, is essentially a rational thing: if I claim to know something, I must be prepared to back up my claim by statingmy reasons for making it;and if my claim is to be upheld, my reasons must begood reasons. Now suppose I know that Q; and let my reasons be conjunctively contained in the proposition that R. Clearly, I must believe that R ;equally clearly, I must know that R . Thus if I know (...) that Q, I know that R. But if I know that R, then I must have my reasons, R' for holding R; and, by the same argument, I mustknow that R'. And if R', then R”; and so on, ad infinitum. (shrink)

The Regress: Knowledge, we like to suppose, is essentially a rational thing: if I claim to know something, I must be prepared to back up my claim by statingmy reasons for making it;and if my claim is to be upheld, my reasons must begood reasons. Now suppose I know that Q; and let my reasons be conjunctively contained in the proposition that R. Clearly, I must believe that R ;equally clearly, I must know that R. Thus if I know that (...) Q, I know that R. But if I know that R, then I must have my reasons, R' for holding R; and, by the same argument, I mustknow that R'. And if R', then R”; and so on, ad infinitum. (shrink)

Book 1 of Euclid's Elements opens with a set of unproved assumptions: definitions, postulates, and ‘common notions’. The common notions are general rules validating deductions that involve the relations of equality and congruence. The attested postulates are five in number, even if a part of the manuscript tradition adds a sixth, almost surely spurious, that in some manuscripts features as the ninth, and last, common notion. The postulates are called αἰτήματα both in the manuscripts of the Elements and in the (...) ancient exegetic tradition. It is not said, however, that this denomination is original, as it coincides with the nomen rei actae associated with the verbal form introducing the postulates themselves. Since antiquity it has been recognized that the postulates naturally split into two groups of quite different character: the first three are rules licensing basic constructions, the fourth and the fifth are assertions stating properties of particular geometric objects. I transcribe postulates 1–3 in the text established by Heiberg : ᾐτήσθω ἀπὸ παντὸς σημείου ἐπὶ πᾶν σημεῖον εὐθεῖαν γραμμὴν ἀγαγεῖνκαὶ πεπερασμένην εὐθεῖαν κατὰ τὸ συνεχὲς ἐπ᾽ εὐθείας ἐκβαλεῖνκαὶ παντὶ κέντρῳ καὶ διαστήματι κύκλον γράϕεσθαι. (shrink)

Arthur Madigan presents a clear, accurate new translation of the third book of Aristotle's Metaphysics, together with two related chapters from the eleventh book. Madigan's accompanying introduction and commentary give detailed guidance to these texts, in which Aristotle sets out the main questions of metaphysics and assesses the main answers to them, and which serve as a useful introduction not just to Aristotle's own work on metaphysics but to classical metaphysics in general.

Arthur Madigan presents a clear, accurate new translation of the third book of Aristotle's Metaphysics, together with two related chapters from the eleventh book. Madigan's accompanying introduction and commentary give detailed guidance to these texts, in which Aristotle sets out the main questions of metaphysics and assesses the main answers to them, and which serve as a useful introduction not just to Aristotle's own work on metaphysics but to classical metaphysics in general.

This is a book about Aristotle's philosophy of language, interpreted in a framework that provides a comprehensive interpretation of Aristotle's metaphysics, philosophy of mind, epistemology and science. The aim of the book is to explicate the description of meaning contained in De Interpretatione and to show the relevance of that theory of meaning to much of the rest of Aristotle's philosophy. In the process Deborah Modrak reveals how that theory of meaning has been much maligned. This is a major reassessment (...) of an underestimated aspect of Aristotle that will be of particular interest to classical philosophers, classicists and historians of psychology and cognitive science. (shrink)

This book examines Aristotle's critical reaction to the mathematical cosmology of Plato's Academy, and traces the aporetic method by which he developed his own ...

For Aristotle, the shape of a physical body is perceptible per se (DA II.6, 418a8-9). As I read his position, shape is thus a causal power, as a physical body can affect our sense organs simply in virtue of possessing it. But this invites a challenge. If shape is an intrinsically powerful property, and indeed an intrinsically perceptible one, then why are the objects of geometrical reasoning, as such, inert and imperceptible? I here address Aristotle’s answer to that problem, focusing (...) on the version of it that he presents in De caelo III.8. I argue that if we grant that Aristotle conceived of the shape of a sensible body as some kind of causal power, then the satisfactory resolution of that challenge pushes us to interpret him as having conceived of it as being, more specifically, an impure power—that is, as a property that is not only intrinsically powerful but also, in some way, intrinsically non-powerful as well. This is a notable result not only insofar as it illuminates Aristotle’s conception of shape but also insofar as it contributes to our knowledge of Aristotle’s theory of dunameis and his ontology more broadly. (shrink)